For our simulation, we take the Dirichlet process prior with \(G={\sf N}(0,1)\)  and \(\delta =1\) . We consider three sample sizes, \(n \in \lbrace 20, 30, 40\rbrace \) , along with three data generating distributions: the normal distribution with mean zero and variance 0.5, the standard Cauchy distribution, and the standard skewed normal distribution [1]} with skewness parameter equal to 1. For each of these scenarios, 5000 data sets are generated and, from each, a 90% prediction interval is extracted. Table REF  reports the estimated coverage probabilities of these prediction intervals. Note how the coverage can be quite low, even for relatively large sample sizes. As one might expect, the coverage probability improves as \(n\)  increases; however, (REF ) effectively involves an infimum over \(\mathsf {P}\) , so the method's performance is determined by the “worst-case distribution,” which can be no better than the results in the Cauchy distribution column in Table REF . Therefore, validity in the sense of (REF ) clearly fails.
<TABLE>